Question

in a 13-sided polygon the sum of 5 angles is 1274 degree four of the eight angles remaining are equal and the other 4 18 degree less than each of the equal angles find the angles

Answer 1

Let's first find the sum of all the angles in a 13-sided polygon using the formula:
Sum of angles = (n-2) * 180 degrees, where n is the number of sides.

For a 13-sided polygon:
Sum of angles = (13-2) * 180 = 11 * 180 = 1980 degrees

Now we know that the sum of 5 angles is 1274 degrees. Let's call the four equal angles x, and the other four angles (which are 18 degrees less than x) y.

We can set up an equation:
5x + 4y = 1274

We also know that the sum of all 13 angles is 1980 degrees. So we can set up another equation:
5x + 4y + 4x + 4(y-18) = 1980

Simplifying this equation:
9x + 6y - 72 = 1980
9x + 6y = 2052
3x + 2y = 684

Now we have two equations with two variables:
5x + 4y = 1274
3x + 2y = 684

You can solve this system of equations using substitution or elimination. Here's one way to do it using elimination:

Multiply the second equation by 2 to get rid of the y variable:
5x + 4y = 1274
6x + 4y = 1368

Subtract the first equation from the second:
x = 94

Now we can substitute x back into one of the equations to find y:
5(94) + 4y = 1274
470 + 4y = 1274
4y = 804
y = 201

So the four equal angles are x = 94 degrees, and the other four angles are y = 201 degrees.

To check that these angles add up correctly, we can verify that:
5x + 4y = 5(94) + 4(201) = 1274
4x + 4(y-18) = 4(94) + 4(183) = 1368
5x + 4y + 4x + 4(y-18) = 9x + 8y - 72 = 1980